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A novel discrete-time repetitive control (RC) scheme is proposed to achieve both the fast convergence rate and the perfect tracking/rejection by updating the input period by period and in the internal of the periods simultaneously. By using the internal model (IM) of the external periodic signals, the classic RC achieves perfect tracking/rejection. Furthermore, a polynomial IM (PIM) is introduced, which only incorporates the dominant frequencies of the external signals. The PIM-based RC is faster than the classic RC because of the fact that the PIM makes the input update in the internal of the periods. However, the PIM-based RC cannot achieve perfect tracking/rejection. Motivated by the fast convergence rate of the PIM-based RC and the perfect tracking/rejection of the classic RC, a new RC scheme is proposed whose IM is the product of the IMs of the classic and PIM-based RCs. As such, the new RC scheme is named as a dual IM (DIM)-based RC. By using the 2-D $H_{\infty }$ theory, it is verified that the DIM-based RC can achieve both the fast convergence rate and perfect tracking/rejection. In addition, by further comparing the classic, PIM-based, and DIM-based RCs in frequency domain, the merits and demerits of the three RCs on the convergence rate and the steady-state performance are deeply discussed. Finally, an experimental apparatus about the rotational system with two dc motors is given to illustrate the advantage of the proposed DIM-based RC. [ABSTRACT FROM AUTHOR]

This note discusses a relationship between the Hankel singular values and reflected zeros of linear systems. Our main result proves that the Hankel singular values of a linear continuous-time system increase (decrease) pointwise when one or more zeros of the transfer function are reflected with respect to the imaginary axis, that is, move from the left-(right-)half to the right-(left-)half of the complex plane. We also derive a similar result for linear discrete-time systems. [ABSTRACT FROM AUTHOR]

On the basis of the relationship of the mth power of a polynomial and its modular form (polynomial whose coefficients are the moduli of the coefficients of that polynomial), we derive a necessary and sufficient condition for the modulus of the mth power of a polynomial for contacting its modular form on the boundary of a disc. Combined with the result about distribution of zeros of analytic function, some new sufficient conditions are derived which give bounds of the absolute values of the roots of a quasi-critical polynomial. These results extend certain earlier similar tests for linear discrete-time systems. Finally, four examples are given to demonstrate the results, Example 2.1 gives a state feedback application, Examples 2.2 and 2.4 deal with r-stability, and Example 2.3 display that our theorems give better results when m increases but at the cost of increasing complexity. [ABSTRACT FROM AUTHOR]